jmc

For any prime number, the sum of its proper divisors is equal to $1$, so a prime number cannot be an abundant number. Therefore, it suffices to check only composite numbers:

$\bullet$ For $4$, $1+2<4$,

$\bullet$ For $6$, $1+2+3=6$,

$\bullet$ For $8$, $1+2+4<8$,

$\bullet$ For $9$, $1+3<9$,

$\bullet$ For $10$, $1+2+5<10$,

$\bullet$ For $12$, $1+2+3+4+6=16>12$.

Thus, the answer is $\boxed{12}$.